GMAT Math : Calculating the height of a right triangle

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Example Question #31 : Right Triangles

Right_triangle

Note: Figure NOT drawn to scale

Refer to the above diagram.

Calculate

Possible Answers:

x=6.66

x=6.72

x=6.52

x = 6.42

Insufficient information is given to calculate .

Correct answer:

x=6.72

Explanation:

The hypotenuse of the large right triangle is

$\sqrt{7^{2} + 24^{2}} = \sqrt{49 + 576} = \sqrt{625} = 25$

The area of the large right triangle is half the product of its base and its height. The base can be any side of the triangle; the height would be the length of the altitude, which is the perpendicular segment from the opposite vertex to that base.

Therefore, the area of the triangle can be calculated as half the product of the legs:

$A = \frac{1}{2} \cdot 7 \cdot 24 = 84$

Or half the product of the hypotenuse and the length of the dashed line.

$A = \frac{1}{2} \cdot 25 \cdot x= \frac{25}{2} x$

To calculate , we can set these expressions equal to each other:

$\frac{25}{2} x = 84$

$\frac{25}{2} x \cdot \frac{2}{25}= 84\cdot \frac{2}{25}$

x=6.72

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Example Question #2 : Calculating The Height Of A Right Triangle

A right triangle has a base of 8 and an area of 24. What is the height of the triangle?

Possible Answers:

Correct answer:

Explanation:

Using the formula for the area of a right triangle, we can plug in the given values and solve for the height of the triangle:

$A=\frac{1}{2}bh$

$24=\frac{1}{2}(8)h$

24=4h

h=6

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Example Question #3 : Calculating The Height Of A Right Triangle

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Triangle ABC is a right triangle with AC=3, AB=4, BC=5 . What is the length of its height ?

Possible Answers:

$\frac{52}{20}$

$\frac{5}{2}$

$\frac{12}{5}$

$\frac{36}{20}$

Correct answer:

$\frac{12}{5}$

Explanation:

The height AE, divides the triangle ABC, in two triangles AEC and AEB with same proportions as the original triangle ABC, this property holds true for any right triangle.

In other words, $\frac{AE}{AB}=\frac{AC}{BC}$ .

Therefore, we can calculate, the length of AE:

$AE= \frac{AC\cdot AB}{BC}= \frac{12}{5}$ .

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Example Question #4 : Calculating The Height Of A Right Triangle

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Triangle ABC is a right triangle with sides AB=8, AC= 6, BC=10 . What is the size of the height ?

Possible Answers:

$\frac{36}{5}$

$\frac{11}{2}$

$\frac{24}{10}$

$\frac{24}{5}$

Correct answer:

$\frac{24}{5}$

Explanation:

As we have previously seen, the height of a right triangle divides a it into two similar triangles with sides of same proportion.

Therefore, we can set up the following equality: $\frac{AE}{AB}=\frac{AC}{BC}$ or $AE=\frac{AC\cdot AB}{BC}$ .

By plugging in the numbers, we get, $\frac{48}{10}$ or $\frac{24}{5}$ .

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Example Question #5 : Calculating The Height Of A Right Triangle

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ABC is a right isosceles triangle, with height . AC=5 , what is the length of the height ?

Possible Answers:

$\frac{5}{2}$

$\frac{5\sqrt{2}}{4}$

$\sqrt{2}$

$\frac{5\sqrt{2}}{2}$

$5\sqrt{2}$

Correct answer:

$\frac{5\sqrt{2}}{2}$

Explanation:

Since here ABC is a isosceles right triangle, its height is half the size of the hypotenuse.

We just need to apply the Pythagorean Theorem to get the length of BC, and divide this length by two.

$BC^{2}=AB^{2}+AC^{2}$ , so $BC=\sqrt{50}$ .

Therefore, $BC=5\sqrt{2}$ and the final answer is $\frac{5\sqrt{2}}{2}$ .

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GMAT Math : Calculating the height of a right triangle

Study concepts, example questions & explanations for GMAT Math

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Example Question #31 : Right Triangles

Example Question #2 : Calculating The Height Of A Right Triangle

Example Question #3 : Calculating The Height Of A Right Triangle

Example Question #4 : Calculating The Height Of A Right Triangle

Example Question #5 : Calculating The Height Of A Right Triangle

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